Hopf Bifurcation Analysis and Existence of Heteroclinic Orbit and Homoclinic Orbit in an Extended Lorenz System
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Hopf Bifurcation Analysis and Existence of Heteroclinic Orbit and Homoclinic Orbit in an Extended Lorenz System Aritra Das1 · Soumya Das2 · Pritha Das2 Accepted: 30 September 2020 © Foundation for Scientific Research and Technological Innovation 2020
Abstract In this paper, we have considered a Lorenz-like model with slight changes in the nonlinear terms. Here we have studied the system dynamics for different range of values of parameters 𝜎, r . The Hopf bifurcation analysis of the system has been done using center manifold theorem for 𝜎 = −1, r > 1 . Phase portraits of solutions of the system are plotted for various system parameters to substantiate the change in dynamics. The bifurcation diagram and the Lyapunov exponent evaluation plots also help to explain the behaviour of the system. Using Fishing principle, we have shown the existence of homoclinic orbit and consequently, observed the existence of homoclinic as well as heteroclinic orbits in the numerical simulation for 𝜎 > 0, r > 1. Keywords Lorenz-like model · Chen system · Hopf bifurcation · Centre manifold theorem · Fishing principle · Homoclinic and heteroclinic orbits
Introduction The nonlinear system of differential equations are studied both from theoretical point of view and from the point of their potential in various areas. Over last two decades, different bifurcation analysis and complex dynamical behaviour have been investigated in many engineering (for example, communication, power systems, etc. [13]) and biological (ecological models, human brain, etc) systems [4]. In 1963, Lorenz [12] studied the nonlinear effect of convection using system of differential equations which are referred as Lorenz model. Using a simplified version of this model, he observed that for some values of the system parameters, the solution trajectories show very complex and unpredictable behaviour. The chaotic behaviour * Pritha Das [email protected] Aritra Das [email protected] Soumya Das [email protected] 1
Department of Physics, Indian Institute of Technology Kanpur, Kanpur, India
2
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, India
13
Vol.:(0123456789)
Differential Equations and Dynamical Systems
originated from the instability of the system for an abrupt atmospheric change. Chen [1, 2, 11] observed some complex dynamics in the system popularly known as Chen system, where using Si’lnikov criterion [14] the existence of chaotic attractor was established. Chen system is referred to as the dual system of the Lorenz system and in the linear part of the Chen model [15] A[aij ], a12 a21 < 0 whereas the Lorenz system satisfies the condition a12 a21 > 0 . Here, we have considered Lorenz like model with slight changes in the nonlinear terms in the following form:
ẋ = 𝜎(−x + ry + ryz) ẏ = −xz + x − y
(1)
ż = −2z + xy where 𝜎, r are parameters and both are not equal to zero. The question we address in this article is the following: How do qualitative changes in the system dynamics occur with a chang
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